%I #33 Feb 11 2024 03:33:39
%S 1,2,1,4,5,1,8,16,8,1,16,44,37,11,1,32,112,134,67,14,1,64,272,424,305,
%T 106,17,1,128,640,1232,1168,584,154,20,1,256,1472,3376,3992,2641,998,
%U 211,23,1,512,3328,8864,12592,10442,5221,1574,277,26,1
%N Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).
%C Row sums are A006190(n+1); diagonal sums are A077939.
%C Inverse is A121575.
%C A generalized Delannoy number triangle.
%C Antidiagonal sums are A002478. - _Philippe Deléham_, Nov 10 2011.
%C From _Peter Bala_, Feb 07 2024: (Start)
%C The following remarks assume the row indexing starts at n = 1.
%C The sequence of row polynomials R(n,x), beginning R(1,x) = 1, R(2,x) = 2 + x, R(3,x) = 4 + 5*x + x^2 , ..., is a strong divisibility sequence of polynomials in the ring Z[x]; that is, for all positive integers n and m, poly_gcd( R(n,x), R(m,x)) = R(gcd(n, m), x) - apply Norfleet (2005), Theorem 3. Consequently, the polynomial sequence {R(n,x): n >= 1} is a divisibility sequence; that is, if n divides m then R(n,x) divides R(m,x) in Z[x]. (End)
%H G. C. Greubel, <a href="/A121574/b121574.txt">Rows n = 0..100 of triangle, flattened</a>
%H M. Norfleet, <a href="https://www.fq.math.ca/Papers1/43-2/paper43-2-12.pdf">Characterization of second-order strong divisibility sequences of polynomials</a>, The Fibonacci Quarterly, 43(2) (2005), 166-169.
%F Number array T(n,k) = Sum_{j=0..n-k} C(k,j)*C(n-j,k)*2^(n-k-j).
%F T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k-1). - _Philippe Deléham_, Nov 10 2011
%F Recurrence for row polynomials (with row indexing starting at n = 1): R(n,x) = (x + 2)*R(n-1,x) + x*R(n-2,x) with R(1,x) = 1 and R(2,x) = x + 2. - _Peter Bala_, Feb 07 2024
%e Triangle begins
%e 1;
%e 2, 1;
%e 4, 5, 1;
%e 8, 16, 8, 1;
%e 16, 44, 37, 11, 1;
%e 32, 112, 134, 67, 14, 1;
%e 64, 272, 424, 305, 106, 17, 1;
%p T:=(n,k)->add(binomial(k,j)*binomial(n-j,k)*2^(n-k-j),j=0..n-k): seq(seq(T(n,k),k=0..n),n=0..9); # _Muniru A Asiru_, Nov 02 2018
%t Table[Sum[Binomial[k, j] Binomial[n-j, k] 2^(n-k-j), {j, 0, n-k}], {n, 0, 15}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Nov 02 2018 *)
%o (PARI) for(n=0,10, for(k=0,n, print1(sum(j=0, n-k, binomial(k, j)* binomial(n-j, k)*2^(n-k-j)), ", "))) \\ _G. C. Greubel_, Nov 02 2018
%o (Magma) [[(&+[ Binomial(k, j)*Binomial(n-j, k)*2^(n-k-j): j in [0..(n-k)]]): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Nov 02 2018
%o (GAP) T:=Flat(List([0..9],n->List([0..n],k->Sum([0..n-k],j->Binomial(k,j)*Binomial(n-j,k)*2^(n-k-j))))); # _Muniru A Asiru_, Nov 02 2018
%Y Cf. Diagonals: A000012, A016789, A080855, A000079, A053220.
%K easy,nonn,tabl
%O 0,2
%A _Paul Barry_, Aug 08 2006